Fundamental mathematical functions that relate angles of a right triangle to ratios of the lengths of its sides, forming the basis for studying periodic phenomena and circular motion.

Trigonometric Functions

Trigonometric functions are essential mathematical tools that emerged from the study of right triangles and have evolved into fundamental concepts used across mathematics, science, and engineering.

Core Functions

The six primary trigonometric functions are:

Sine (sin): The ratio of the opposite side to the hypotenuse
Cosine (cos): The ratio of the adjacent side to the hypotenuse
Tangent (tan): The ratio of the opposite side to the adjacent side
Cosecant (csc): The reciprocal of sine
Secant (sec): The reciprocal of cosine
Cotangent (cot): The reciprocal of tangent

Properties and Relationships

Fundamental Identities

sin²θ + cos²θ = 1 (The Pythagorean Identity)
tan θ = sin θ / cos θ
periodic functions at regular intervals

Domain and Range

Each function has specific domains where they are defined and ranges of possible values:

sin(x): Domain = all real numbers, Range = [-1, 1]
cos(x): Domain = all real numbers, Range = [-1, 1]
tan(x): Domain = all real numbers except π/2 + πn, Range = all real numbers

Applications

Trigonometric functions find extensive applications in:

Historical Development

The development of trigonometric functions spans multiple civilizations:

Ancient Babylonian mathematics astronomical calculations
Greek geometry chord tables
Indian mathematics sine tables
Islamic golden age scholars' systematic development

Modern Extensions

Modern understanding extends these functions beyond right triangles to:

Visualization Tools

Understanding these functions is often aided by:

The study of trigonometric functions continues to be essential in modern mathematics and its applications, forming a bridge between geometric and analytical thinking.